Approximation by $\alpha$-Bernstein-Schurer operator

Nursel Çetin

doi:10.15672/hujms.626905

Araştırma Makalesi

Approximation by $\alpha$-Bernstein-Schurer operator

Yıl 2021, Cilt: 50 Sayı: 3, 732 - 743, 07.06.2021

Nursel Çetin

https://doi.org/10.15672/hujms.626905

Cited By: 2

Öz

In this paper, we introduce a new family of generalized Bernstein-Schurer operators and investigate some approximation properties of these operators. We obtain a uniform approximation result using the well-known Korovkin theorem and give the degree of approximation via second modulus of smoothness. Also, we present Voronovskaya and Grüss-Voronovskaya type results for these operators.

Anahtar Kelimeler

Bernstein-Schurer operators, α-Bernstein operator, Modulus of continuity, Voronovskaya type theorem, Grüss-Voronovskaya type theorem

Kaynakça

[1] T. Acar and A. Kajla, Degree of approximation for bivariate generalized Bernstein type operators, Results Math. 73 Article number: 79, 2018.
[2] T. Acar, A.M. Acu and N. Manav, Approximation of functions by genuine Bernstein- Durrmeyer type operators, J. Math. Inequal. 12 (4), 975–987, 2018.
[3] A.M. Acu, H. Gonska and I. Raşa, Grüss-type and Ostrowski-type inequalities in approximation theory, Ukrainian Math. J. 63 (6), 843–864, 2011.
[4] A. Aral and H. Erbay, Parametric generalization of Baskakov operators, Math. Com- mun. 24, 119–131, 2019.
[5] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applica- tions, Walter de Gruyter, Berlin-New York, 1994.
[6] S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Commun. Kharkov Math. Soc. 13, 1–2, 1912/1913.
[7] Q. Cai and X. Xu, Shape-preserving properties of a new family of generalized Bern- stein operators, J. Inequal. Appl. 2018, Article number: 241, 2018.
[8] D. Cárdenas-Morales, P. Garrancho and I. Raşa, Bernstein-type operators which pre- serve polynomials, Comput. Math. Appl. 62 (1), 158–163, 2011.
[9] X. Chen, J. Tan and Z. Liu, Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl. 450, 244–261, 2017.
[10] N. Çetin, Approximation and geometric properties of complex -Bernstein operator, Results Math. 74, Article number: 40, 2019.
[11] N. Çetin and V.A. Radu, Approximation by generalized Bernstein-Stancu operators, Turkish J. Math. 43, 2032–2048, 2019.
[12] R.A. DeVore and G.G. Lorentz, Constructive Approximation, Springer, Berlin, 1993.
[13] J.L. Durrmeyer, Une formule d’inversion de la transformée de Laplace-Applications à la théorie des moments, Thèse de 3e cycle, Faculté des Sciences, Université de Paris, 1967.
[14] S.G. Gal and H. Gonska, Grüss and Grüss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables, Jaen J. Approx. 7 (1), 97– 122, 2015.
[15] H. Gonska and G. Tachev, Grüss-type inequalities for positive linear operators with second order moduli, Mat. Vesnik, 63 (4), 247–252, 2011.
[16] G. Grüss, Uber das maximum des absoluten betrages von $\frac{1}{b-a}\int\limits_{a}^{b}f\left( x\right) g\left( x\right) dx-\frac{1}{\left( b-a\right) ^{2}}\int\limits_{a}^{b}f\left( x\right)dx\int\limits_{a}^{b}g\left( x\right) dx,$, Math. Z. 39, 215–226, 1935.
[17] A. Kajla and D. Miclăuş, Blending Type Approximation by GBS Operators of Gener- alized Bernstein–Durrmeyer Type, Results Math. 73 Article number: 1, 2018.
[18] L.V. Kantorovich, Sur certains développements suivant les polynômes de la forme de S. Bernstein, I, II, C.R.Acad.URSS, 563–568, 595–600, 1930.
[19] A. Lupaş, A q-analogue of the Bernstein operator, Seminar on Numerical and Statis- tical Calculus, Babeş-Bolyai University, 9, 85–92, 1987.
[20] S.A. Mohiuddine, T. Acar and A. Alotaibi, Construction of a new family of Bernstein- Kantorovich operators, Math. Methods Appl. Sci. 40, 7749–7759, 2017.
[21] F. Schurer, Linear positive operators in approximation theory, Math. Inst. Techn. Univ. Delft Report, 1962.
[22] D.D. Stancu, Asupra unei generalizări a polinoamelor lui Bernstein, Studia Universi- tatis Babeş-Bolyai, Ser. Math.-Phys. 14 (2), 31–45, 1969 (in Romanian).
[23] D.D. Stancu, Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20 (2), 211–229, 1983.
[24] J. Szabados, On a quasi-interpolating Bernstein operator, J. Approx. Theory, 196, 1–12, 2015.
[25] E. Voronovskaja, Détermination de la forme asymptotique d’approximation des fonc- tions parles polynômes de S. Bernstein, Dokl. Akad. Nauk SSSR, 79–85, 1932.

Yıl 2021, Cilt: 50 Sayı: 3, 732 - 743, 07.06.2021

Nursel Çetin

https://doi.org/10.15672/hujms.626905

Cited By: 2

Öz

Kaynakça

[1] T. Acar and A. Kajla, Degree of approximation for bivariate generalized Bernstein type operators, Results Math. 73 Article number: 79, 2018.
[2] T. Acar, A.M. Acu and N. Manav, Approximation of functions by genuine Bernstein- Durrmeyer type operators, J. Math. Inequal. 12 (4), 975–987, 2018.
[3] A.M. Acu, H. Gonska and I. Raşa, Grüss-type and Ostrowski-type inequalities in approximation theory, Ukrainian Math. J. 63 (6), 843–864, 2011.
[4] A. Aral and H. Erbay, Parametric generalization of Baskakov operators, Math. Com- mun. 24, 119–131, 2019.
[5] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applica- tions, Walter de Gruyter, Berlin-New York, 1994.
[6] S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Commun. Kharkov Math. Soc. 13, 1–2, 1912/1913.
[7] Q. Cai and X. Xu, Shape-preserving properties of a new family of generalized Bern- stein operators, J. Inequal. Appl. 2018, Article number: 241, 2018.
[8] D. Cárdenas-Morales, P. Garrancho and I. Raşa, Bernstein-type operators which pre- serve polynomials, Comput. Math. Appl. 62 (1), 158–163, 2011.
[9] X. Chen, J. Tan and Z. Liu, Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl. 450, 244–261, 2017.
[10] N. Çetin, Approximation and geometric properties of complex -Bernstein operator, Results Math. 74, Article number: 40, 2019.
[11] N. Çetin and V.A. Radu, Approximation by generalized Bernstein-Stancu operators, Turkish J. Math. 43, 2032–2048, 2019.
[12] R.A. DeVore and G.G. Lorentz, Constructive Approximation, Springer, Berlin, 1993.
[13] J.L. Durrmeyer, Une formule d’inversion de la transformée de Laplace-Applications à la théorie des moments, Thèse de 3e cycle, Faculté des Sciences, Université de Paris, 1967.
[14] S.G. Gal and H. Gonska, Grüss and Grüss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables, Jaen J. Approx. 7 (1), 97– 122, 2015.
[15] H. Gonska and G. Tachev, Grüss-type inequalities for positive linear operators with second order moduli, Mat. Vesnik, 63 (4), 247–252, 2011.
[16] G. Grüss, Uber das maximum des absoluten betrages von $\frac{1}{b-a}\int\limits_{a}^{b}f\left( x\right) g\left( x\right) dx-\frac{1}{\left( b-a\right) ^{2}}\int\limits_{a}^{b}f\left( x\right)dx\int\limits_{a}^{b}g\left( x\right) dx,$, Math. Z. 39, 215–226, 1935.
[17] A. Kajla and D. Miclăuş, Blending Type Approximation by GBS Operators of Gener- alized Bernstein–Durrmeyer Type, Results Math. 73 Article number: 1, 2018.
[18] L.V. Kantorovich, Sur certains développements suivant les polynômes de la forme de S. Bernstein, I, II, C.R.Acad.URSS, 563–568, 595–600, 1930.
[19] A. Lupaş, A q-analogue of the Bernstein operator, Seminar on Numerical and Statis- tical Calculus, Babeş-Bolyai University, 9, 85–92, 1987.
[20] S.A. Mohiuddine, T. Acar and A. Alotaibi, Construction of a new family of Bernstein- Kantorovich operators, Math. Methods Appl. Sci. 40, 7749–7759, 2017.
[21] F. Schurer, Linear positive operators in approximation theory, Math. Inst. Techn. Univ. Delft Report, 1962.
[22] D.D. Stancu, Asupra unei generalizări a polinoamelor lui Bernstein, Studia Universi- tatis Babeş-Bolyai, Ser. Math.-Phys. 14 (2), 31–45, 1969 (in Romanian).
[23] D.D. Stancu, Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20 (2), 211–229, 1983.
[24] J. Szabados, On a quasi-interpolating Bernstein operator, J. Approx. Theory, 196, 1–12, 2015.
[25] E. Voronovskaja, Détermination de la forme asymptotique d’approximation des fonc- tions parles polynômes de S. Bernstein, Dokl. Akad. Nauk SSSR, 79–85, 1932.

Toplam 25 adet kaynakça vardır.

Ayrıntılar

Birincil Dil	İngilizce
Konular	Matematik
Bölüm	Matematik
Yazarlar	Nursel Çetin 0000-0003-3771-6523
Yayımlanma Tarihi	7 Haziran 2021
Yayımlandığı Sayı	Yıl 2021 Cilt: 50 Sayı: 3

Kaynak Göster

APA	Çetin, N. (2021). Approximation by $\alpha$-Bernstein-Schurer operator. Hacettepe Journal of Mathematics and Statistics, 50(3), 732-743. https://doi.org/10.15672/hujms.626905
AMA	Çetin N. Approximation by $\alpha$-Bernstein-Schurer operator. Hacettepe Journal of Mathematics and Statistics. Haziran 2021;50(3):732-743. doi:10.15672/hujms.626905
Chicago	Çetin, Nursel. “Approximation by $\alpha$-Bernstein-Schurer Operator”. Hacettepe Journal of Mathematics and Statistics 50, sy. 3 (Haziran 2021): 732-43. https://doi.org/10.15672/hujms.626905.
EndNote	Çetin N (01 Haziran 2021) Approximation by $\alpha$-Bernstein-Schurer operator. Hacettepe Journal of Mathematics and Statistics 50 3 732–743.
IEEE	N. Çetin, “Approximation by $\alpha$-Bernstein-Schurer operator”, Hacettepe Journal of Mathematics and Statistics, c. 50, sy. 3, ss. 732–743, 2021, doi: 10.15672/hujms.626905.
ISNAD	Çetin, Nursel. “Approximation by $\alpha$-Bernstein-Schurer Operator”. Hacettepe Journal of Mathematics and Statistics 50/3 (Haziran 2021), 732-743. https://doi.org/10.15672/hujms.626905.
JAMA	Çetin N. Approximation by $\alpha$-Bernstein-Schurer operator. Hacettepe Journal of Mathematics and Statistics. 2021;50:732–743.
MLA	Çetin, Nursel. “Approximation by $\alpha$-Bernstein-Schurer Operator”. Hacettepe Journal of Mathematics and Statistics, c. 50, sy. 3, 2021, ss. 732-43, doi:10.15672/hujms.626905.
Vancouver	Çetin N. Approximation by $\alpha$-Bernstein-Schurer operator. Hacettepe Journal of Mathematics and Statistics. 2021;50(3):732-43.

Hacettepe Journal of Mathematics and Statistics

Approximation by $\alpha$-Bernstein-Schurer operator

Öz

Anahtar Kelimeler

Kaynakça

Öz

Kaynakça

Ayrıntılar

Kaynak Göster

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Mathematics

https://doi.org/10.3390/math10183227